   Chapter 14, Problem 46RE

Chapter
Section
Textbook Problem

Find the directional derivative of f at the given point in the indicated direction.46. f ( x , y , z ) = x 2 y + x 1 + z , (1,2, 3), in the direction of v = 2i+,j − 2k

To determine

To find: The directional derivative of the function f(x,y,z)=x2y+x1+z at the point (1,2,3) in the direction of the vector v=2i+j2k .

Explanation

Given:

The functionis f(x,y,z)=x2y+x1+z .

Result used:

“The directional derivative of the function f(x,y,z) at f(x0,y0,z0) in the direction of unit vector u=a,b,c is Duf(x,y,z)=f(x,y,z)u , where f(x,y,z)=fx,fy,fz=fxi+fyj+fzk .”

Calculation:

The directional derivative is defined as, Duf(x,y,z)=f(x,y,z)u (1)

The value of f(x,y,z) is computed as follows.

f(x,y,z)=fx,fy,fz=x(x2y+x1+z),y(x2y+x1+z),z(x2y+x1+z)=(2xy+1+z),(x2),(x21+z)

Thus, the value of f(x,y,z)=(2xy+1+z),(x2),(x21+z)

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